few_shots: Example:
Problem/activity:
For each pair of numbers, determine if the values are equivalent. Explain your reasoning for each one.
a.
$$-0.25$$
and
$$\frac{1}{-4}$$
b.
$$-\left ( \frac{1}{4} \right )$$
and
$$\frac{-1}{-4}$$
c.
$$\frac{-1}{4}$$
and
$$-\frac{1}{4}$$


Topic options: 
A. Operations & Algebra: students learn to solve problems using algebraic thinking and operations such as addition, subtraction, multiplication, and division. They may learn to identify and explain arithmetic patterns, evaluate or manipulate numerical expressions, and reason with equations and inequalities.
B. Functions: students learn to define, use, and evaluate functions to model relationships between quantities.
C. Counting & Cardinality: students learn to know number names and the count sequence, count to tell the number of objects, and compare numbers.
D. Number & Operations - Fractions: students learn to understand fractions as numbers and may work with them using addition, subtraction, multiplication, and division.
E. Number & Operations in Base Ten: students learn to work with the base-ten system and build place value understanding.
F. Statistics & Probability: students learn to analyze and produce data distributions, and build understanding of univariate and bivariate data. They learn to interpret data, make statistical inferences, justify conclusions, understand rules of probability, and use probability to make decisions.
G. Ratios & Proportional Relationships: students learn to recognize, describe, represent, and reason with ratios, rates, proportional relationships, and percent.
H. Measurement & Data: students learn to work with data and measure attributes such as time, money, length, area, and volume. They may learn to compare measurements to operations and convert between different units of measure.
I. Number Systems and Quantity: students learn to understand the complex number system and reason quantitatively.
J. Geometry: students learn to classify geometric figures by their properties, and understand and compare the relationships between them. Students develop and use formulas to compute lengths, areas, and volumes, and they use transformations to generate new shapes and compare existing ones. Geometry can be studied with and without coordinates.
K. Modeling: students learn to choose and use appropriate mathematics and statistics to analyze real-world empirical situations, improve decisions, and report on their conclusions and the reasoning behind them.


Thought: This problem teaches students different ways of writing numbers that express the same value. This helps students better understand the complex number system, which includes rational numbers.
Answer: I

Now, assign the following problem or activity to one or more relevant topics it teaches.
shot_info: fl_problem_001097

7
options: Ratios & Proportional Relationships: students learn to recognize, describe, represent, and reason with ratios, rates, proportional relationships, and percent.

Measurement & Data: students learn to work with data and measure attributes such as time, money, length, area, and volume. They may learn to compare measurements to operations and convert between different units of measure.

Operations & Algebra: students learn to solve problems using algebraic thinking and operations such as addition, subtraction, multiplication, and division. They may learn to identify and explain arithmetic patterns, evaluate or manipulate numerical expressions, and reason with equations and inequalities.

Statistics & Probability: students learn to analyze and produce data distributions, and build understanding of univariate and bivariate data. They learn to interpret data, make statistical inferences, justify conclusions, understand rules of probability, and use probability to make decisions.

Modeling: students learn to choose and use appropriate mathematics and statistics to analyze real-world empirical situations, improve decisions, and report on their conclusions and the reasoning behind them.

Geometry: students learn to classify geometric figures by their properties, and understand and compare the relationships between them. Students develop and use formulas to compute lengths, areas, and volumes, and they use transformations to generate new shapes and compare existing ones. Geometry can be studied with and without coordinates.

Number Systems and Quantity: students learn to understand the complex number system and reason quantitatively.

Number & Operations - Fractions: students learn to understand fractions as numbers and may work with them using addition, subtraction, multiplication, and division.

Number & Operations in Base Ten: students learn to work with the base-ten system and build place value understanding.

Counting & Cardinality: students learn to know number names and the count sequence, count to tell the number of objects, and compare numbers.

Functions: students learn to define, use, and evaluate functions to model relationships between quantities.
problem_activity: Activity
In this task, students use the constant of proportionality they estimated in the previous task to calculate circumferences of circles given the diameter and vice versa. The purpose is to reinforce the proportional relationship between circumference
\(c\)
and diameter
\(d\)
and use the formula
\(c=kd\)
, where
\(k\)
is about 3 or 3.1 (the value agreed upon in the previous task).
If there is not sufficient time to allow students to work through all of the calculations, consider dividing up the work, assigning one circle to each student or group of students. It is important to save enough time for the discussion about the meaning and value of
\(\pi\)
at the end of this activity.
Before students begin the task, a digital applet found at
http://ggbm.at/H8UuD96V
can be used to reinforce the constant of proportionality.
Launch
Ask students to label these measurements given in the table on the picture of the circles. Instruct students to use the constant of proportionality the class estimated in the previous activity. Give students quiet work time followed by whole-class discussion.
Student Facing
Here are five circles. One measurement for each circle is given in the table.
[IMG]
Use the constant of proportionality estimated in the previous activity to complete the table.
|          | diameter (cm) | circumference (cm) |
|----------|---------------|--------------------|
| circle A |               |                    |
| circle B |               |                    |
| circle C |               |                    |
| circle D |               |                    |
| circle E |               |                    |
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Student Response.
Student Facing
Are you ready for more?
The circumference of Earth is approximately 40,000 km. If you made a circle of wire around the globe, that is only 10 meters (0.01 km) longer than the circumference of the globe, could a flea, a mouse, or even a person creep under it?
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Extension Student Response.
Anticipated Misconceptions
Some students may multiply the circumference by the constant of proportionality instead of dividing by it. Prompt them to consider whether the diameter can be longer than the circumference of a circle.
Some students may struggle to divide by 3.1 if that is the constant of proportionality decided on in the previous lesson. Ask these students if they could use an easier number as their constant, and allow them to divide by 3 instead. Then ask them how their answer would have changed if they divided by 3.1.
Activity Synthesis
Display the table from the task statement for all to see. Ask students to share their answers and write these in the table, resolving any discrepancies.
Ask students if they noticed any connections between the tables and the graphs from the previous activity. Students will have used the equation
\(C=kd\)
, possibly without realizing it, to fill in the missing information so it is important for them to make this connection. To reinforce this connection, add the points from the table they just completed to the graph from the previous activity. Point out that the constant of proportionality appears in the table as the unit rate in the row with diameter 1 cm.
Ask if students have heard of the number
pi
or seen the symbol
\(\pi\)
. Define
\(\pi\)
as the exact value of the constant of proportionality for this relationship. Explain that the exact value of
\(\pi\)
is a decimal with infinitely many digits and no repeating pattern, so an approximation is often used. Frequently used approximations for
\(\pi\)
include
\(\frac{22} 7\)
, 3.14, and 3.14159, but none of these are exactly equal to
\(\pi\)
.


correct_option_index: 0

5